Dynamic Analysis of a Combined Spiral Bevel Gear and Planetary Gear Set in a Bucket Elevator with High Power Density

Abstract

A combined spiral bevel gear and planetary gear set was designed for bucket elevators to reduce the system’s complexity and improve the system’s efficiency. The dynamic model of a combined spiral bevel gear and a 2K-H planetary gear train was developed by using the lumped mass method, where the time-varying mesh stiffness and comprehensive mesh error were introduced. Then, the equations of motion of the system were solved with the Newmark numerical integration method, and the dynamic transmission error and mesh force of the gear pairs were obtained. The influences of different input rotational speeds and power on the dynamic response of the system were analyzed. The dynamic responses of the bearings between the bevel gears, planet carrier, and box were calculated. It was found that the dynamic mesh force gradually decreased with an increase in speed, and it increased with an increase in power. For low-speed and heavy-load transmission systems, the influence of power on the mesh force was more significant. It could also be found that the efficiency of the transmission system composed of a spiral bevel gear and a planetary gear was improved compared with that of the traditional system. Finally, the effectiveness of the dynamic model was validated through a comparison of the test signals. The results of the analysis in this study can be used to guide the design of vibration and noise reduction.

1. Introduction

At present, there are many problems in the working processes of bucket elevators, such as drastic rises in temperature in low-power conditions and high energy consumption in high-power conditions. Sometimes, hydraulic coupling is adopted to keep a higher power, which increases the complexity of the whole system. To solve the above problems, a drive-system-coupled permanent magnet motor and gear train are proposed. In this system, a permanent magnet motor is the power drive, and a combined spiral bevel gear and planetary gear train are used as the transmission system. This system integrates the advantages of permanent magnet motors and gear drives, reduces the system complexity, and makes the structure more compact. Moreover, spiral bevel gears have a high load-carrying capacity, long service life, no root-cut problems, and an easily adjustable transmission ratio [1]. The design of vibration and noise reduction with a high power density is more important in this proposed system. To improve the performance and efficiency of the product, it is necessary to carry out dynamic optimization of this spiral bevel gear–planetary gear transmission system.

Xu et al. [2] developed a dynamic bending–torsion model of a spiral bevel gear pair by using the lumped mass method. The time-varying stiffness, backlash, bearing clearance, and static transmission errors were considered in this model. The influences of the mesh frequency, gear backlash, and load coefficient on the gear system were analyzed. Cheng and Tang [3,4], respectively, established dynamic models of a gear pair and analyzed the effects of design parameters on the vibration characteristics. A dynamic model considering various excitations was introduced [5,6,7]. Wang et al. analyzed the effect of excitation frequency on dynamic characteristics. Zeng et al. proposed an integral transformation method that could convert a frequency-domain response signal into excitation. Gou, Dong, and Xiang [8,9,10] established a coupling dynamic model of gear transmission with the finite element method and the lumped mass method. The dynamic characteristics, the time-varying load, and the mesh stiffness were calculated. Yavuz et al. [11,12] constructed a nonlinear time-varying dynamic model of a transmission system considering the backlash of the tooth and time-varying mesh stiffness. Lafi et al. [13,14] constructed a dynamic model of a differential bevel gear set. The free and forced vibration response of the gear system was studied. The Hertz contact stiffness, bending stiffness, compression stiffness, and shear stiffness were computed. Their inherent frequencies and vibration types were also analyzed. Cao et al. [15] developed a frictional dynamic model of a spiral bevel gear and analyzed the effect of dynamics on lubrication performance and fatigue life.

A dynamic model of a coupled box vibration and 2K-H planetary gear transmission system was constructed by He et al. [16]. Gu et al. [17] established a lumped mass model considering the planetary wheel position error and analyzed the effect of position error on the quasi-static and dynamic load distribution between planets. Velex et al. [18] established a dynamic model for multi-tooth meshing gear transmission. The relationship between dynamic meshing forces and gearing errors was theoretically analyzed. Yang et al. [19] analyzed the dynamic response of cyclic and random loads on a planetary gear transmission system. Jian et al. [20] analyzed the thermal elastohydrodynamic lubrication characteristics of a gear system under dynamic loads. Wang et al. [21] derived the influence of the meshing phase on a planetary transmission system and validated these laws through a simulation analysis. Wang and Tang et al. [22,23] constructed a dynamic model that considered the integrated error, backlash nonlinearity, and the effects of input speed, damping ratio, tooth backlash, and integrated meshing error. Kahnamouei et al. [24] established a dynamic model considering the elastic deformation of a tooth ring, the backlash nonlinearity, and the variation in the meshing stiffness based on the finite element method. The closed-form amplitude–frequency relations and parametric instability bandwidth expressions were derived from Wang’s model [25]. Yang et al. [26] studied the effects of carrier and bearing flexibility on the meshing stiffness and investigated the effect of crack expansion at the root of the sun gear and the ring on the meshing stiffness, load-carrying capacity, and NVH characteristics with a sensitivity analysis model. Xiang and Wang et al. [27,28] established a dynamic model of a multistage planetary gear train. The effect of gear backlash on the dynamic characteristics of the system was analyzed. Karray et al. [29] use the lumped mass method to establish a mathematical model of a spiral bevel gear and two-stage planetary gear, and they carried out a modal analysis on it.

This paper mainly studies the dynamic characteristics of a transmission system involving the combination of a spiral bevel gear and planetary gear train. In this paper, the dynamic model of the combined spiral bevel gears and planetary gear transmission set is established in a coupled permanent magnet motor and gear drive system. The time-varying meshing stiffness and mesh error are considered in order to improve the accuracy of this model. By using the Newmark method, dynamic mesh forces are analyzed, and the influences of input speed and load on the mesh forces are also discussed. Finally, the simulated dynamic response signal is compared with the measured vibration signal in the test to verify the validity of the dynamic model. The research results can be a useful guide for vibration and noise reduction design.

2. Dynamic Modeling

Figure 1a shows the dynamic model of a spiral bevel gear–planetary gear train. The whole system consists of a pair of bevel gears and a set of 2K-H planetary gears. Figure 1b shows the physical model of the transmission system.

Figure 1. Dynamic model and physical model of the transmission system. (a) Dynamic model of the transmission system. (b) Physical model of the transmission system.

In the bevel–planetary gear transmission system, a coupling dynamic model containing bending–torsional–axial effects was established for the spiral bevel gears. Since all planet gears are spur gears, a bending–torsional coupling model is established for the planetary gear train. The axial force generated by the bevel gear is transmitted to the bearings through the shaft segment, then to the box, and finally to the frame through the fixed bolt. Therefore, the planetary gears are not subject to axial force.

2.1. Dynamic Equation of a Spiral Bevel Gear

The right-angle coordinate system was established by taking the intersection point of the two axes of the drive and driven gears as the coordinate origin. The axis of the drive bevel gear was the X axis, and the axis of the large bevel gear was the Z axis. Both the large and small bevel gears had four degrees of freedom (DOFs), these were two bending directions, the torsional direction of the rotation axes, and the axial direction.

As shown in Figure 2, the normal dynamic load of the mesh gears and its component forces along each coordinate direction could be obtained according to the mechanical relations of the drive bevel gear.

$$\left\{\begin{array}{c}{F}_{\mathrm{n}}=k_{\mathrm{m}}(\mathrm{t}){\delta }_{12}(t)+c_{\mathrm{m}}{\dot{\delta }}_{12}(\mathrm{t})\\ F_x=F_{\mathrm{n}}(\sin {\alpha }_{\mathrm{n}}\cos {\delta }_1+\cos {\alpha }_{\mathrm{n}}\sin {\beta }_1\sin {\delta }_1)\\ F_y=F_{\mathrm{n}}(\sin {\alpha }_{\mathrm{n}}\sin {\delta }_1-\cos {\alpha }_{\mathrm{n}}\sin {\beta }_1\cos {\delta }_1)\\ F_z=F_{\mathrm{n}}\cos {\alpha }_{\mathrm{n}}\cos {\beta }_1\end{array}\right.$$

(1)

Figure 2. Force analysis of the drive bevel gear’s teeth.

Here, k_m(t) represents the time-varying mesh stiffness; c_m is the mesh damping; α_n is the normal pressure angle; β₁ is the helix angle at the midpoint of the small bevel gear; δ₁ represents the pitch cone angle of the small bevel gear.

We assumed that

$$\left\{\begin{array}{c}{a}_1=\sin {\alpha }_{\mathrm{n}}\cos {\delta }_1+\cos {\alpha }_{\mathrm{n}}\sin {\beta }_1\sin {\delta }_1\\ a_2=\sin {\alpha }_{\mathrm{n}}\sin {\delta }_1-\cos {\alpha }_{\mathrm{n}}\sin {\beta }_1\cos {\delta }_1\\ a_3=\cos {\alpha }_{\mathrm{n}}\cos {\beta }_1\end{array}\right.$$

(2)

The generalized displacement coordinates of the bevel gear pair can be expressed as follows:

$${\mathit{q}}_{\mathrm{pg}}={{\displaystyle \{x_{\mathrm{p}},y_{\mathrm{p}},z_{\mathrm{p}},{\theta }_{\mathrm{xp}},x_{\mathrm{g}},y_{\mathrm{g}},z_{\mathrm{g}},{\theta }_{\mathrm{zg}}\}}}^T$$

(3)

where x_i, y_i, and z_i (i = p, g) are the displacements of the drive gear (p) or the large gear (g) in three directions. θ_xp and θ_zg represent the torsion angles of the gears around the X axis and Z axis, respectively. By projecting the large and small bevel gear vibration displacements along the line of action, the relative displacement can be obtained as follows:

$$\begin{array}{ll}{\delta }_{12}(\mathrm{t})=& a_1(x_{\mathrm{p}}-x_{\mathrm{g}})+a_2(y_{\mathrm{p}}-y_{\mathrm{g}})+a_3(z_{\mathrm{p}}-z_{\mathrm{g}})+\\ & r_{\mathrm{p}}{\theta }_{\mathrm{xp}}-r_{\mathrm{g}}{\theta }_{\mathrm{zg}}-e_{\mathrm{m}}(\mathrm{t})\end{array}$$

(4)

where e_m(t) represents the normal mesh error of the gear pair; r_p and r_g are the equivalent base radii at the gear reference points.

According to Newton’s second law, the dynamic differential equations for the spiral bevel gear pair can be expressed as follows:

$$\left\{\begin{array}{c}{m}_{\mathrm{p}}{\ddot{x}}_{\mathrm{p}}+c_{1\mathrm{x}}{\dot{x}}_{\mathrm{p}}+k_{1\mathrm{x}}{x}_{\mathrm{p}}=-F_x\\ m_{\mathrm{p}}{\ddot{y}}_{\mathrm{p}}+c_{1\mathrm{y}}{\dot{y}}_{\mathrm{p}}+k_{1\mathrm{y}}{y}_{\mathrm{p}}=-F_y\\ m_{\mathrm{p}}{\ddot{z}}_{\mathrm{p}}+c_{1\mathrm{z}}{\dot{z}}_{\mathrm{p}}+k_{1\mathrm{z}}{z}_{\mathrm{p}}=-F_z\\ \begin{array}{c}\frac{I_{\mathrm{xp}}}{{{\displaystyle r}}_{\mathrm{p}}^2}{\ddot{u}}_{\mathrm{xp}}=\frac{T_1}{r_{\mathrm{p}}}-F_{\mathrm{n}}\\ m_{\mathrm{g}}{\ddot{x}}_{\mathrm{g}}+c_{2\mathrm{x}}{\dot{x}}_{\mathrm{g}}+k_{2\mathrm{x}}{x}_{\mathrm{g}}=F_x\\ m_{\mathrm{g}}{\ddot{y}}_{\mathrm{g}}+c_{2\mathrm{y}}{\dot{y}}_{\mathrm{g}}+k_{2\mathrm{y}}{y}_{\mathrm{g}}=F_y\\ \begin{array}{c}{m}_{\mathrm{g}}{\ddot{z}}_{\mathrm{g}}+c_{2\mathrm{z}}{\dot{z}}_{\mathrm{g}}+k_{2\mathrm{z}}{z}_{\mathrm{g}}=F_z\\ \frac{I_{\mathrm{zg}}}{{{\displaystyle r}}_{\mathrm{g}}^2}{\ddot{u}}_{\mathrm{zg}}-k_{\mathrm{u}}{u}_{\mathrm{s}}=-\frac{T_2}{r_{{}_{\mathrm{g}}}}+F_{\mathrm{n}}\end{array}\end{array}\end{array}\right.$$

(5)

where m_p and m_g are the large and small bevel gear masses, respectively; I_xp and I_zg represent the rotational inertias of the gears around the X axis and Z axis; T₁ is the input torque; T₂ is the output torque; k_ij (i = 1, 2; j = x, y, z) represents the stiffness coefficient of each part in the direction of coordinate axis; c_ij (i = 1, 2; j = x, y, z) is the damping coefficient of each part in the direction of coordinate axis; k_u means the torsional stiffness of the shaft segment. u_ij (i = x, z; j = p, g) represents the displacement of the gear in the torsional direction, and u_s is the torsional line displacement of the sun gear.

2.2. Dynamic Model of the Planetary Gear Transmission

As shown in Figure 1, the sun gear is represented by s, r is the ring gear, c is the planet carrier, and the planet gear is represented by n. O_iX_iY_i is the dynamic coordinate system of each part. x_i, y_i, and θ_i (i = s, c, r, n₁, n₂, n₃) indicate the translational and torsional elastic deformations of each part. k_ix and k_iy (i = s, c, r, n₁, n₂, n₃) represent the support stiffness of each bearing in the X and Y directions. k_iu (i = s, c, r) is the torsional stiffness of each part. θ_ni (i = 1, 2, 3) is the mounting angle of the planet gears.

For the subsequent calculations, the displacement of the gear mesh pair along the line of action is denoted by u_i = r_i × θ_i, where r_i (i = s, c, r, n₁, n₂, n₃) is the base radius of each gear, and u_i (i = s, c, r, n₁, n₂, n₃) represents the torsional line displacement of each part.

Projecting the mesh pair between the sun gear and the planetary gears along the direction of the line of action, the relative deformation is expressed by δ_sni (i = 1, 2, 3). The projected relative deformation of the ring–planet gear mesh pair along the line of action is represented by δ_rni (i = 1, 2, 3). The projected relative deformations between the planetary gear and the planet carrier along the x_ni and y_ni axes are δ_cnxi and δ_cnyi.

$${\delta }_{\mathrm{sni}}=x_{\mathrm{s}}\sin {\theta }_{\mathrm{sni}}-y_{\mathrm{s}}\cos {\theta }_{\mathrm{sni}}+u_{\mathrm{s}}+x_{\mathrm{ni}}\sin \alpha -y_{\mathrm{ni}}\cos \alpha +u_{\mathrm{ni}}-e_{\mathrm{sn}}(\mathrm{t})$$

(6)

$${\delta }_{\mathrm{rni}}=x_{\mathrm{r}}\sin {\theta }_{\mathrm{rni}}-y_{\mathrm{r}}\cos {\theta }_{\mathrm{rni}}+u_r+x_{\mathrm{ni}}\sin \alpha -y_{\mathrm{ni}}\cos \alpha -u_{\mathrm{ni}}-e_{\mathrm{rn}}(\mathrm{t})$$

(7)

$${\delta }_{\mathrm{cnxi}}=x_c\sin {\theta }_{\mathrm{ni}}+y_c\cos {\theta }_{\mathrm{ni}}-x_{\mathrm{ni}}$$

(8)

$${\delta }_{\mathrm{cnyi}}=x_c\sin {\theta }_{\mathrm{ni}}-y_c\cos {\theta }_{\mathrm{ni}}-y_{\mathrm{ni}}+u_c$$

(9)

where θ_sni = α + θ_ni(i = 1, 2, 3), θ_rni = α − θ_ni(i = 1, 2, 3). θ_sni and θ_rni are the angles between the line of action of the sun and planetary gears and the X axis, as well as the angles between the line of action between the planetary wheel and internal ring gear and the X axis, respectively. θ_ni is the installation angle of the ith planetary wheel. α is the pressure angle, and e_sn(t) and e_rn(t) are comprehensive mesh errors.

According to Newton’s second law, the dynamic differential equations for the planetary transmission system can be obtained as follows [30].

The dynamic model of the sun gear is represented by

$$\left\{\begin{array}{c}{m}_{\mathrm{s}}{\ddot{x}}_{\mathrm{s}}+k_{\mathrm{s}}{x}_{\mathrm{s}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{sn}}{\delta }_{\mathrm{sni}}\sin {\theta }_{\mathrm{sn}}{}_{\mathrm{i}}}=0\\ m_{\mathrm{s}}{\ddot{y}}_{\mathrm{s}}+k_{\mathrm{s}}{y}_{\mathrm{s}}-{\displaystyle \sum _{i=1}^3{k}_{\mathrm{sn}}{\delta }_{\mathrm{sni}}\cos {\theta }_{\mathrm{sni}}}=0\\ \frac{I_{\mathrm{s}}}{r_{\mathrm{s}}^2}{\ddot{u}}_{\mathrm{s}}+k_{\mathrm{su}}{u}_{\mathrm{s}}+k_{\mathrm{u}}{u}_{\mathrm{s}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{sn}}{\delta }_{\mathrm{sn}}{}_{\mathrm{i}}}=\frac{T_{\mathrm{p}}}{r_{\mathrm{s}}}\end{array}\right.$$

(10)

The dynamic differential equations of the planetary gear are

$$\left\{\begin{array}{c}{m}_{\mathrm{n}}{\ddot{x}}_{\mathrm{ni}}-k_{\mathrm{cn}}{\delta }_{\mathrm{cnxi}}-k_{\mathrm{sn}}{\delta }_{\mathrm{sni}}\sin \alpha +k_{\mathrm{rn}}{\delta }_{\mathrm{rni}}\sin \alpha =0\\ m_{\mathrm{n}}{\ddot{y}}_{\mathrm{ni}}-k_{\mathrm{cn}}{\delta }_{\mathrm{cnyi}}-k_{\mathrm{sn}}{\delta }_{\mathrm{sni}}\cos \alpha -k_{\mathrm{rn}}{\delta }_{\mathrm{rni}}\cos \alpha =0\\ \frac{I_{\mathrm{ni}}}{r_{\mathrm{n}}^2}{\ddot{u}}_{\mathrm{ni}}-k_{\mathrm{rn}}{\delta }_{\mathrm{rni}}+k_{\mathrm{sn}}{\delta }_{\mathrm{sni}}=0\end{array}\right.$$

(11)

The dynamic equations of the ring gear can be obtained as follows:

$$\left\{\begin{array}{c}{m}_{\mathrm{r}}{\ddot{x}}_{\mathrm{r}}+k_{\mathrm{r}}{x}_{\mathrm{r}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{rn}}{\delta }_{\mathrm{rni}}\sin {\theta }_{\mathrm{rni}}=0}\\ m_{\mathrm{r}}{\ddot{y}}_{\mathrm{r}}+k_{\mathrm{r}}{y}_{\mathrm{r}}-{\displaystyle \sum _{i=1}^3{k}_{\mathrm{rn}}{\delta }_{\mathrm{rni}}\cos {\theta }_{\mathrm{rni}}=0}\\ \frac{I_{\mathrm{r}}}{r_{\mathrm{r}}^2}{\ddot{u}}_{\mathrm{r}}+k_{\mathrm{ru}}{u}_{\mathrm{r}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{rn}}{\delta }_{\mathrm{rni}}=0}\end{array}\right.$$

(12)

The dynamic differential equations of the planetary carrier are expressed as follows:

$$\left\{\begin{array}{c}{m}_{\mathrm{c}}{\ddot{x}}_{\mathrm{c}}+k_{\mathrm{c}}{x}_{\mathrm{c}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{cn}}{\delta }_{\mathrm{cnxi}}\cos {\theta }_{\mathrm{ni}}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{cn}}{\delta }_{\mathrm{cnyi}}\sin {\theta }_{\mathrm{ni}}}=0\\ m_{\mathrm{c}}{\ddot{y}}_{\mathrm{c}}+k_{\mathrm{c}}{y}_{\mathrm{c}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{cn}}{\delta }_{\mathrm{cnxi}}\sin {\theta }_{\mathrm{ni}}}-{\displaystyle \sum _{i=1}^3{k}_{\mathrm{cn}}{\delta }_{\mathrm{cnyi}}\cos {\theta }_{\mathrm{ni}}}=0\\ \frac{I_{\mathrm{c}}}{{{\displaystyle r}}_{\mathrm{c}}^2}{\ddot{u}}_{\mathrm{c}}+k_{\mathrm{cu}}{u}_{\mathrm{c}}+{\displaystyle \sum _{i=1}^3{k}_{\mathrm{cn}}{\delta }_{\mathrm{cnyi}}}=\frac{T_{\mathrm{g}}}{r_{\mathrm{c}}}\end{array}\right.$$

(13)

where m_i (i = s, c, r, n) is the mass of the sun gear, planet, ring gear, and carrier. I_i (i = s, c, r, n) represents the moment of inertia of the sun gear, planet gear, ring gear, and planet carrier. k_in (i = s, r) is the mesh stiffness between each mesh pair. k_cn is the support stiffness between the carrier and planetary gear. k_i (i = s, c, r) is the support stiffnesses of the sun, the ring gear, and the planet carrier. r_c is the radial distance from the axis’ center of the planet to the geometric center of the planet carrier. T_i (i = p, g) represents the input and output torque of the planetary transmission system.

The general dynamic differential equation of the combined spiral bevel gear and planetary transmission system can be obtained by combining Equation (5) with Equation (13).

$$\mathit{M}\ddot{\mathit{q}}+\mathit{C}[\dot{\mathit{q}}-\dot{e}(\mathrm{t})]+\mathit{K}[\mathit{q}-e(\mathrm{t})]=\mathit{F}$$

(14)

where M is the whole mass matrix of the system; C is the whole damping matrix; K is the whole stiffness matrix; F is the excitation vector; q is the generalized displacement vector of the system; e(t) represents the comprehensive mesh errors.

3. Dynamic Analysis of the System

3.1. Parameters of the Model

It is assumed that the planetary gears are uniformly distributed, and each planet has the same inertia, mass, mesh stiffness, and support stiffness. The support stiffnesses of the sun, the planet carrier, and the ring gear in the X and Y directions are the same. The drive system is applied to a bucket elevator and is the main drive part of the bucket elevator’s drive system. The bucket elevator is mainly used for vertical transportation of concrete, so it needs to maintain a low speed and high torque. The drive power of the bucket elevator is 90–280 kW, and the output speed is 16–30 r/min. According to the product design manual, the input power of the drive system is 200 kW, the input speed is 300 r/min, and the input torque is 6367 N·m [31]. The basic parameters of the gear transmission are shown in Table 1.

Table 1. Basic parameters of the gear transmission.

Parameter	Small Bevel Gear	Large Bevel Gear	Sun	Planet	Inner Ring Gear	Planet Carrier
Modulus	10.85	10.85	4.5	4.5	4.5	-
Number of teeth	16	57	23	35	93	-
Equivalent diameter (m)	0.18	0.56	0.1	0.16	0.42	0.34
Pressure angle (°)	20	20	20	20	20	-
Helix angle (°)	30	30	0	0	0	-
Mass (kg)	2.14	118.73	7.38	5.20	36.71	242.75
Moment of inertia (kg·m2)	0.16	5.41	0.01	0.29	1.83	7.03
Support stiffness (N/m)	2 × 108	2 × 108	1.18 × 109	5 × 107	7.38 × 109	8.63 × 108
Torsional stiffness (N/m)	2.9 × 108	2.9 × 108	3.47 × 109	-	83.4 × 109	3.64 × 108

The meshing frequency of each part of the system can be calculated according to the basic parameters. The first-order rotation frequency of the motor rotor is 50 Hz, which is the same as the power frequency. The first-order meshing frequency of the bevel gear meshing is 60 Hz, and the first-order meshing frequency of the planetary gear system meshing is 25 Hz. It can be found that the second- and third-order meshing frequencies of each component have a corresponding relationship of multiples with the first-order meshing frequency.

The mesh stiffness k_m(t) of the spiral bevel gear is calculated with the finite element method. Based on the idea of the energy slice method and according to [32], the time-varying meshing stiffness of a single pair of teeth is calculated when the coincidence ratio of the planetary transmission is 1. The time-varying mesh stiffness of the internal and external gear pairs can be given in Figure 3. The figure shows the change in meshing stiffness in a cycle. The dimensionless period is the ratio of the time that a pair of teeth pass from entering the meshing position to the meshing period. The number of gears participating in the meshing changes alternately with the rotation angle of the gear during meshing, so the meshing stiffness of the gear also changes periodically with time. When two pairs of gears are engaged in the transmission, the meshing stiffness can be regarded as the parallel connection of two pairs of teeth, that is, the superposition of the comprehensive stiffness of a single pair of teeth. The stiffness of double-tooth meshing is greater than that of single-tooth meshing. In the transition region of single- and double-tooth meshing, the gear meshing stiffness has a sudden change.

Figure 3. Time-varying mesh stiffness.

3.2. Dynamic Transmission Error

The dynamic displacement of the system can be obtained by solving the dynamic model of the transmission system with Newmark’s integration method. Further, the dynamic transmission error of the gear pair along the direction of the line of action is calculated with Equation (15) [33]. We define the sun gear and planetary gear 1 as the external mesh pair 1, and we define the sun gear and planetary gear 2 as the external mesh pair 2. The rest of the mesh pairs can be defined in the same manner.

$$DTE=\mathit{V}\cdot \mathit{q}$$

(15)

where V is the displacement transform vector; q is the displacement vector of the mesh unit.

Figure 4 shows the dynamic transmission error of each gear pair. Here, three meshing cycles of the result after reaching a steady state are selected for the analysis. Because the speed of the bevel gear is higher than that of the planetary gear train, the dynamic transmission error of the bevel gear pair is obviously larger than that of the planetary gear train. For the planetary gear pair, the variation tendency of the dynamic transmission error is almost the same. By comparing the results with those in [34], it can be found that the peak value of the dynamic transmission error of each gear pair is within a reasonable range.

Figure 4. Dynamic transmission error of the gear pairs. (a) The spiral bevel gear pair. (b) The mesh pair of the sun and the planet. (c) The mesh pair of the inner ring gear and the planet.

3.3. Dynamic Mesh Force

After obtaining the dynamic transmission error of a gear pair, the dynamic mesh force of the gear pair can be obtained with Equation (16).

$$F_{\mathrm{ij}}=k_{\mathrm{ij}}(\mathrm{t})\cdot {\delta }_{\mathrm{ij}}(\mathrm{t})$$

(16)

where k_ij(t) (i = s, r; j = n) represents the mesh stiffness of the ith gear pair; δ_ij(t) (i = s, r; j = n) represents the relative displacement between gear pairs.

Figure 5 shows the dynamic mesh force of each gear pair. The maximum value of the dynamic mesh force between the bevel gear pairs is 3.85 × 10⁴ N. The average value of the mesh force is around 3.84 × 10⁴ N. Because the planets are mounted at different angles, this will lead to a phase difference in the results of the analysis. In Figure 5b, it can be observed that the maximum dynamic mesh force of the first external mesh gear pair is 8.76 × 10⁴ N, and the average mesh force is around 8.55 × 10⁴ N.

Figure 5. Dynamic response of the mesh force. (a) Mesh force of the spiral bevel gear pair. (b) Mesh force between the sun and the planet. (c) Mesh force between the inner ring gear and the planet.

The maximum value of the dynamic mesh force of the second outer gear pair is 8.75 × 10⁴ N, and the average mesh force is around 8.53 × 10⁴ N. The maximum dynamic mesh force of the third external gear pair is 8.77 × 10⁴ N, and the average value is around 8.56 × 10⁴ N.

In Figure 5c, it can be observed that the maximum dynamic mesh force of the first internal gear pair is 8.76 × 10⁴ N, and the average mesh force is around 8.55 × 10⁴ N. The maximum dynamic mesh force of the second internal gear pair is 8.76 × 10⁴ N, and the average force is about 8.53 × 10⁴ N. In the transmission system, due to the different mesh phases of each planet, the mesh forces are slightly different in their peaks and fluctuation amplitudes. The peak value of the dynamic meshing force of gear pairs at all levels is less than the force calculated with the empirical formula, and the peak value of the dynamic meshing force is within a reasonable range [35].

3.4. Influence of Rotational Speed on Dynamic Mesh Force

Figure 6 shows the influence of different rotational speeds on the first external mesh pair. The effects of rotating speed on the internal meshing pair force are similar. When the rotating speed reaches 280 r/min, the maximum external mesh force is equal to 100.22 kN; when the input speed is 320 r/min, the maximum external mesh force is equal to 76.53 kN. As mentioned above, the mesh force decreases as the rotating speed increases. When the power remains unchanged, the input speed increases, the input torque decreases, and the decrease in torque makes the mesh force decrease.

Figure 6. Effects of different rotational speeds on the external mesh force.

3.5. Influence of Different Loads on Dynamic Mesh Force

Figure 7 shows the influence of different power levels on the first external mesh gear pair. The effects of power on the internal meshing pair force are similar. When the input power reaches 180 kW, the maximum external mesh force is equal to 70.54 kN; when the input power is 220 kW, the maximum external engagement force is equal to 104.94 kN. As mentioned above, the mesh force increases as the power increases. When the speed is constant, the input power increases, the input torque increases, and the increases in the torque make the mesh force increase.

Figure 7. Effects of different power values on the external mesh force.

4. Discussion and Validation

To verify the validity of the dynamic model of the spiral bevel gear–planetary gear transmission system, a vibration test device for a bucket elevator drive system was established. As shown in Figure 8, a coupling, adapter plate, and torque meter were used to connect a tractor motor with a bucket elevator drive system. The test equipment and the types used are shown in Table 2. The dynamic characteristics of the drive system were studied by adding measurement points to the box. As shown in Figure 9, set measuring point 2 and measuring point 4 were placed on the box to measure vibration. Measuring points 2 and 4 corresponded to bearings 2 and 4, respectively. In Figure 1a, k2 and c2 marked on the left side of the large bevel gear represent bearing 2, and k4 and c4 marked on the right side of the planetary gear train represent bearing 4.

Figure 8. Test platform for the dynamic characteristics of a bucket elevator drive system. 1: tractor motor; 2: adapter plate; 3: coupling; 4: torque meter; 5: Drive system.

Table 2. Test equipment.

Test Equipment	Type	Test Equipment	Type
Tractor motor	TYCPT 400S-20	Coupling	SWC285
Adapter plate	MH3S170H22.4AT	Torque meter	JN338-20000
Vibration measurement system	B&K vibration measuring system (including one-way vibration measuring probe)	Sensor	B&K speed sensor

Figure 9. Sampling position of the measuring points.

When carrying out a no-load test on the drive system, the vibration speed data at the measuring point were collected every 5 min before reaching the rated speed, and the vibration speed data were collected every 15 min after reaching the rated speed. The data acquisition instrument was connected with the vibration measurement system and a monitor. Then, the speed without load is controlled by the speed sensor, and the vibration response of each measuring point was measured with a vibration probe. Finally, the vibration data were recorded with the data acquisition instrument.

The vibration velocity of the bearing was obtained by solving the dynamic differential equations for the second bearing and the fourth bearing. The vibration velocity spectrum of the bearing could be obtained with a fast Fourier transform. Figure 10 indicates the variations in the vibration velocity of the measurement points in the frequency domain. Comparing the simulation results with the experimental results, it could be found that the overall change trend of the simulation data was more consistent with the trend of the measured data, with a slight difference at the peak. As shown in Table 3, the differences between the test and simulation data were 3.80% and 8.84%, which verified the validity of the simulation model. In conjunction with Section 2.1, the peak values at point 2 in the test and simulation data were mainly concentrated at 50, 60, and 120 Hz, which were the same as the first-order excitation frequency of the motor rotor and the first-order and second-order excitation frequencies of the bevel gear. Therefore, it can be considered that the vibration of the spiral bevel gear was mainly affected by the meshing frequency and the motor excitation. The peak values at point 4 in the test and simulation data were mainly concentrated at 50 and 100 Hz, which were the same as the first-order excitation frequency of the bevel gear and the second-order excitation frequency of the planetary gear. Therefore, it could be considered that the vibration of the planetary gear carrier was mainly caused by the first-order excitation frequency of the planetary gear set and the second-order excitation frequency of the spiral bevel gear.

Figure 10. Vibration velocities in the tests and simulations.

Table 3. Comparison of the vibration velocities between the tests and simulations.

Measurement Point	Test Peak (μm/s)	Simulation Peak (μm/s)	Error (%)
Point 1	514.26	533.80	3.80
Point 2	260.55	283.58	8.84

The efficiency of the whole system was tested through the test platform that was built. After starting the tractor motor first, the system was operated at the rated power and rated voltage without load. When the bearing temperature and oil pool temperature became stable, the measurement and recording of the input power were implemented once. Then, to connect the drive system with the tractor motor, the system was run at the rated voltage and rated power without load. When the bearing and oil pool became stable, the input power was measured and recorded again, and the efficiency of the drive system was obtained. The test results show that the transmission efficiency of the whole system including the motor was 94.1%. The calculated value of the motor efficiency was 96.7%. The transmission system included the spiral bevel gear and the planetary gear train, and the total efficiency of the transmission system was 96.6%. Therefore, the calculated efficiency of the drive system including the motor reached 93.4%, which can be obtained with Equation (17). Due to friction and loss during transmission, the actual efficiency of the system was smaller than the calculated value, but the error value was also within the acceptable range. The transmission device of a traditional bucket elevator includes a hydraulic coupler and reducer. The reducer has two stages, and the transmission efficiency is 95%. The efficiency of the hydraulic coupler is 95%. The total efficiency of a traditional transmission system including the motor is only 87.3%. It could be found that the improved transmission system eliminated the hydraulic coupler, reduced the loss of efficiency, and introduced spiral bevel gears and planetary gear trains with higher transmission efficiency, so the transmission efficiency was significantly improved.

$$\eta ={\eta }_1\cdot {\eta }_2$$

(17)

where η₁ is the transmission efficiency of the spiral bevel gear; η₂ is the transmission efficiency of the planetary gear train; η represents the total efficiency of the transmission system.

5. Conclusions

The dynamic model of a combined spiral bevel gear and planetary gear train set was developed based on the lumped mass method. The effects of time-varying gear mesh stiffness and errors of the mesh pairs were considered in this model. The dynamic transmission errors and dynamic mesh forces of each gear pair were obtained with Newmark’s integration method in the system.

The vibration of the spiral bevel gear in the transmission system was mainly caused by the motor excitation and the gear mesh frequency. The vibration of the inner ring of the planetary gear system was mainly caused by the first-order mesh frequency of the planetary gear and the second-order mesh frequency of the bevel gear. The influence of the input torque made the dynamic transmission error of the spiral bevel gear pair significantly larger than that of the planetary gear system. By analyzing the influences of different input rotation speeds and power on the dynamic mesh force, we found that the dynamic mesh force gradually decreased with the increase in speed, and it increased with the increase in power. For a low-speed heavy-load transmission system, the influence of power on the mesh force was more significant. It could be found that the efficiency of the transmission system was improved according to a comparison of test and simulation data. The validity of the dynamic model of the spiral bevel gear–planetary gear train was validated by comparing the vibration results at the same measurement points in the experiment and simulation. In future research, the dynamic characteristics of the drive system can be further studied, and a permanent magnet synchronous motor and drive system can be introduced to study an electromechanical coupling system.